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601-hw-vi

601-hw-vi - MATH 601 AUTUMN 2008 Additional homework VI...

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MATH 601, AUTUMN 2008 Additional homework VI, November 6 PROBLEMS 1. Let an endomorphism T of a finite-dimensional vector space V such that have n distinct eigenvalues, where n = dim V , and let an endomorphism S of V commute with T . Prove that S is diagonalizable. 2. Let T be an endomorphism of a two-dimensional real vector space V such that det T < 0. Prove that T is diagonalizable. 3. Is the same conclusion as in Problem 2 true under the weaker assuption det T 0? 4. Given an integer n 0, let V be the ( n +1)-dimensional space of all polynomials f of the real variable x with deg f n , and let D : V V be the derivative operator with Df = f . Find a concise description of e tD : V V . 5. Find an explicit formula for the solutions x ( t ) , y ( t )) to the system of ordinary differential equations ˙ x = 3 x - 9 y , ˙ y = x - 3 y with any given initial conditions x (0) = a , y (0) = b , a, b C . 6. Let an endomorphism T : V V of a vector space V satisfy the condition f ( T ) = 0 for some given polynomial f . Prove that every eigenvalue λ of T

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601-hw-vi - MATH 601 AUTUMN 2008 Additional homework VI...

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